Stephane Legault, Mitch Moroz and Travis Ewanyk had the other goals for the Oil Kings, who have a 1-0 series lead in the Eastern Conference second-round matchup.
"They're a very tough team," said Kulda. "Today they showed us that it's not just good luck (that they're here), it's hard to play against them. It will not be an easy series for us."
Kyle Becker scored the lone goal for the Tigers, who had trouble at times producing scoring chances.
"There were stretches that were fine, obviously we weren't able to generate a lot of offense," said Tigers' bench boss Shaun Clouston. "Last series against Saskatoon, we were able to generate on the transition and tonight that didn't show up as much. Our success against Saskatoon, we really played as a group. Tonight we didn't work back to the puck enough, we got spread out a little bit and Edmonton was able to pick passes off."
Kulda opened up the scoring at 15:53 of the first period after taking the puck off the end boards at the edge of the crease and jamming it between the legs of Medicine Hat netminder Cam Lanigan.
Early in the second period, Edmonton goaltender Laurent Brossoit robbed Boston Leier from the high slot by flashing the leather on a shot labeled for the top corner to keep the Oil Kings ahead.
Moments later, Legault doubled the Oil Kings' lead off a feed from Michael St. Croix with what turned out to be the eventual winner.
Moroz's goal, which was a wrap-around at 5:54 of the third period, pushed Edmonton ahead 3-0.
Becker put an end to Brossoit's bid at his third shutout of the post-season as he put the Tigers on the scoreboard at 7:30 with a shot just under the crossbar.
Ewanyk iced the game with a high-slot wrister to the glove side at 17:33.
Brossoit stopped 21 shots in net for Edmonton as Lanigan made 29 saves for Medicine Hat.
"We'll make some adjustments but I thought we did a really good job up-ice," said Edmonton head coach Derek Laxdal. "They didn't get a lot of penetration when we were on the penalty kill, and when (Brossoit) had to be strong, he was strong. We're trying to play a more uptempo game on the forecheck and I thought our guys did a really good job on that."
The Oil Kings went 0 for 4 on the power play while the Tigers went 0 for 5.
Game 2 is scheduled for Sunday in Edmonton at Rexall Place.